Before the Page time: maximum entanglements or the return of the monster?

The information loss paradox is an unresolved problem that contains the essential nature of the quantum theory of gravity Hawking:1976ra . If we assume semi-classical general relativity and the unitarity of quantum mechanics, then inconsistent conclusions follow Yeom:2008qw . Accordingly, we must further clarify the five assumptions as follows Susskind:1993if :

• – A1.

  Unitarity: There is no loss of information from black hole formation to complete evaporation.

• – A2.

  Local quantum field theory: The computation of Hawking radiation is consistent.

• – A3.

  General relativity: Classical description is sufficient unless the curvature is larger than the Planck scale.

• – A4.

  Bekenstein-Hawking entropy as statistical entropy: The Boltzmann entropy carried by a black hole is expressed by $A/4$, where $A$ denotes the horizon area.

• – A5.

  The existence of an information observer: There exists an observer who can read and distinguish information from Hawking radiation.

These five assumptions cannot be consistent. In essence, either there exists an observer who can detect the duplication of information Yeom:2008qw or the laws of entanglement entropy are violated Almheiri:2012rt .

The majority of candidate resolutions of the information loss paradox can be categorized by denying at least one of the five assumptions Sasaki:2014spa . These are outlined below:

• – R1.

  Violation of unitarity: Information can be lost Hawking:1976ra ; Unruh:1995gn .

• – R2.

  Violation of the local quantum field theory: non-local interactions exist between inside and outside the horizon Giddings:2012gc .

• – R3.

  Violation of general relativity: There may exist a membrane Thorne:1986iy , a fuzzball Mathur:2005zp , or a firewall Almheiri:2012rt approximal to the horizon.

• – R4.

  Existence of monsters: Information can be retained by a small or large remnant Chen:2014jwq . Such a remnant can be a type of baby universe Hwang:2010gc .

• – R5.

  Absence of an information observer: No (semi-classical) observer exists to read information from Hawking radiation, and therefore, information is effectively lost Lee:2015rwa .

In our discussion, our arguments strongly rely on the result of the Page curve Page:1993df . According to Page:1993df , when we consider a pure and random bipartite system ($n$ and $m$ are the number of states for each subsystem and $1\ll m\leq n$), the entanglement entropy, $S$, between two subsystems is approximated

Therefore, small subsystems are always approximately in the thermal state because the entanglement entropy is essentially the same as the Boltzmann entropy Lloyd:1988cn , i.e.,

This indicates that we cannot obtain distinguishable information from Hawking radiation if the number of radiation states is sufficiently smaller than states of inside the black hole. If we apply this to an evaporating black hole, then information can only be obtained after the Page time, i.e., the time when two subsystems have the same number of states ($m=n$). After the Page time, the number of states outside the horizon becomes larger than the number inside, and therefore, it is possible to read the information outside the horizon. Accordingly, it is not surprising that many contradictory situations occur after the Page time Yeom:2008qw ; Almheiri:2012rt , which is the main focus of many papers.

Contrastingly, in this paper, we examine events before the Page time. Before the Page time, the number of states inside a black hole is dominated. As Hawking particles are emitted, the entanglements between the inside and outside of a black hole linearly increase Hwang:2016otg . Furthermore, the increasing entanglement entropy takes on a maximum value and, accordingly, the thermal nature is preserved. The problem remains, however, of whether this phenomenon is a natural consequence. The assumptions for this are outlined below (Fig. 1) Hwang:2017yxp :

• – A1/A3

  : A pair-creation of particles is a unitary process near the horizon and the background is nothing special near the horizon. Hence, the pair-creation process is described by the creation of a separable state⁵⁵5In this paper, we consider the creation of a particle-antiparticle pair (or the combination of many particles as a separable state), where separable state refers to no entanglement between the previous particles (black hole or radiation) and the created particles; however, of course, there must exist entanglements among the created particles..

• – A2

  : All interactions are local. Thus, after the antiparticles fall into the black hole, they only interact with the degrees of freedom inside the black hole (and vice versa). Since there are no interactions between inside and outside the horizon, to preserve maximum entanglements (following the Page argument), the Hawking particles themselves must be maximally entangled.

• – A1/A4

  : After particle emission, the black hole area decreases. Hence, there should exist a unitary annihilation process between antiparticles and the internal degrees of freedom. In other words, there exists a process consisting of the annihilation of a separable state.

If these three key points are repeated, then one can explain the emission of particles, at least, before the Page time Hwang:2017yxp .

In this paper, we examine whether these three key points are consistent with each other. If Hawking radiation is dominated by free fields, then it is not surprising that Hawking radiation can be described by maximally entangled pairs. However, if the field is interacting, processes more complicated than a single particle and single antiparticle processes can exist. In addition, even though a particle-antiparticle pair is created with maximum entanglements, if two pairs interact with each other, then it is possible for the maximum entanglement assumption to be violated; accordingly, the two particles and two antiparticles are totally randomized. In turn, this raises the problem of whether the Page curve is valid before the Page time. By using spin-1/2 toy models Hwang:2016otg ; Hwang:2017yxp ; Bae:2019niw , we demonstrate the possibility of this alongside various interpretations.

This paper is organized as follows. In Sec. II, we discuss several preliminary topics; in Sec. III, we examine whether the field is interacting, from which the bias from the maximum entanglements can be accumulated; in Sec. IV, we discuss the potential implications of our conclusion; and in Sec. VI, we summarize the paper and discuss possible future research directions.

We used numerical simulations to estimate the entanglement entropy of the half of a randomly mixed two-, four-, six-, or eight-particle system, the results of which are shown in Fig. 5. The left figures of Fig. 5 consist of the entanglement entropy curves for a two-, four-, six-, and eight-particle system. By repeating the numerical simulations, many curves were superposed. For each trial, we evaluated the half-way entanglement entropy for a two-, four-, six-, and eight-particle system⁸⁸8Therefore, the horizontal axis of right figures do not mean the time direction; these mean trials in order to evaluate statistical tendencies.. This was shown in the right figures of Fig. 5, from which it is evident that an average value $\langle S(A|B)\rangle$ exists and the result is diversely distributed with a given standard deviation $\sigma$. The numerical results are summarized in Table 1.

Indeed, as the number of particles increases, the standard deviation decreases and the entropy reaches a maximum. This means that if maximally entangled pairs interact with many particles, then the entanglement entropy of the half of the total system will almost be the maximum possible value. Alternatively, if maximally entangled particles interact only with a small number of particles, the deviation from the maximum entanglement entropy cannot be negligible. Our results concerning the expectation values of entanglement entropy are consistent with analytic computations by Bianchi:2019stn ; to be sure, the standard deviations by Bianchi:2019stn are lightly different because the analytic form is limited for cases with a large number of particles.

Using this numerical observation, we can now estimate the bias from the originally expected Page curve before the Page time. This is approximated as follows: two processes exist, where one is the creation of a maximally entangled pair and the other is the creation of randomly mixed four-particles. The probability of the first process is $p_{1}$ and that of the second process is $p_{2}$, where $p_{1}\gg p_{2}$ and $p_{1}+p_{2}=1$. From the first process, for each step, the number of radiation states $\log m$ increases by $\log 2$; simultaneously, the entanglement entropy inside and outside also increases by $\log 2$. Therefore, if $p_{2}=0$, then $\log m-S\simeq 0$ and the original Page curve is confirmed. However, from the second process, for each step, the number of radiation states increases by $2\log 2$, because two particles are emitted; the entanglement entropy increases by $\mathcal{D}<2\log 2$, where the numerical value of $\mathcal{D}$ is $0.9222$ from the previous simulation. By changing $p_{2}$, we can see the bias from the original Page curve (Fig. 6).

From the numerical simulations (Fig. 6), it is evident that the bias from the original Page curve is indeed proportional to $\log m$. If the total number of events is $\mathcal{N}$, the number of events of the two-particle process is ${\mathcal{N}}_{1}=p_{1}\mathcal{N}$, whereas that of the four-particle process is ${\mathcal{N}}_{2}=p_{2}\mathcal{N}$. For each process of ${\mathcal{N}}_{1}$, the entanglement entropy increases by $\log 2$; for each process of ${\mathcal{N}}_{2}$, the entanglement entropy increases by $\mathcal{D}$. The maximum value of the increase in entanglement entropy is $\log m={\mathcal{N}}_{1}\log 2+2{\mathcal{N}}_{2}\log 2=\mathcal{N}(p_{1}+2p_{2})\log 2$. Alternatively, the true entanglement entropy change is ${\mathcal{N}}_{1}\log 2+{\mathcal{N}}_{2}\mathcal{D}=\mathcal{N}(p_{1}\log 2+p_{2}\mathcal{D})$. Therefore, the expected emission of information is

This expectation is confirmed by comparing pink lines (analytic estimation) and green curves (numerical computation) in Fig. 6. If $p_{2}$ is too small, then the bias from the Page curve is negligible. However, if $p_{2}$ is not zero and $\log m$ is sufficiently large (i.e., of the order of $\log m\sim p_{2}^{-1}$), there should be a detectable bias.

From the previous discussion, we can conclude that a bias exists from the Page curve if pair-created particles are not maximally entangled with their counterparts. Even though the probability is not high, if a black hole size is large enough, then the accumulated bias will not be negligible. This implies that the bias from the exact thermal state before the Page time becomes

Accordingly, there must exist a distinguishable effect from Hawking radiation.

Then what is the physical meaning of this conclusion? In this section, we illustrate possible interpretations and consequences.

We have considered three alternative possibilities. First, we considered the possibility that information is carried out before the Page time via Hawking radiation. However, this results inconsistency; information might be duplicated. Second, we considered firewalls or non-local effects. Third, we considered remnants or monster-like object formation, where the Boltzmann entropy exceeds its areal entropy.

Regarding the second possibility, we need a mechanism to realize non-local interactions. In this context, it is worthwhile to compare our results with the recent discussion in the string community about the entanglement entropy issue. Based on the quantum extremal surface technique Engelhardt:2014gca , one could obtain the entanglement entropy of an evaporating black hole Almheiri:2019hni . According to their computation, before the Page time, the entanglement entropy monotonically increases because the dominant contribution comes from the original Hawking’s saddle, while after the Page time, the dominant contribution comes from a new saddle that can be interpreted as the replica wormholes Almheiri:2019qdq . However, these saddles will contribute after the Page time, and it is hard to believe that they can contribute in the early stage of black hole evaporations.

In addition to this, still it is unclear to understand the detailed process to carry out information from the black hole. The difficulty comes from the fact that the entanglement entropy computation is a path integral of the density matrices, while the authentic quantum mechanics is about a path integral of the quantum states Chen:2021jzx . Because of this problem, even though one computes the entanglement entropy on the Euclidean geometry, it is very unnatural to analytically continue to the Lorentzian signatures. There have been some trials to interpret in the Lorentzian signatures Marolf:2020rpm , but it is still too premature to conclude the final interpretation.

Probably, the third option might be the most conservative approach, but we need more justifications. In this regards, although it is a minor opinion, some researchers proposed an idea that if we do not keep the central dogma Buoninfante:2021ijy , i.e., the assumption A4 (Bekenstein-Hawking entropy as statistical entropy), one may overcome the difficulties of the information loss paradox. As proposed in Buoninfante:2021ijy , it is conceptually possible to derive a monster-like state that has more statistical entropy than its areal entropy. Recent computations from Euclidean path integral approach Chen:2021jzx also indicates the same direction. Our paper proposes a new mechanism that can derive a monster-like state; the interesting new point of ours is that the monster-like state can be derived even before the Page time.

Therefore, although it is not conclusive, our approach sheds some interesting intuitions to the recent discussion of the string community. For example, if the monster-like state construction is allowed via a matter field with self-interactions as we expected in our paper, the contribution from the islands can be effective even before the Page time, because usually one believes that the monster-like behavior is one signature of the existence of the island contribution, although this is not quite consistent to the so-called mainstream picture of the information loss paradox. Further discussion is beyond the scope of this paper; we left them for future research topics.

In this paper, we investigated the particle emission process from a black hole using spin-1/2 toy models. It is worthwhile to mention that our analysis have some limitations. We considered spin-1/2 particles with finite degrees of freedom, while quantum fields have infinite degrees of freedom. However, our analysis will shed some lights to the information loss paradox, because we regard that the quantum state has a finite entropy as well as a finite degrees of freedom. Therefore, in terms of qubits and their entanglements, the same relationship must be applicable not only for particles but also for quantum states of fields.

In doing so, the traditional consensus was that, before the Page time, Hawking radiation is generated by a particle-antiparticle pair creation, where two particles are maximally entangled and form a separable state. Alternatively, we considered a situation where two maximally entangled pairs interact with each other; or where a separable four-particle system is created from a vacuum (Sec. II). Accordingly, it was assumed that the infalling two particles are randomized with the outgoing two particles, which implies that the incoming particles do not satisfy the maximum entanglement condition with the outgoing particles.

To the authors’ knowledge, there is no fundamental reason to abandon this possibility, although the probability can be less dominant. As time passes, the bias from the original Page curve before the Page time is accumulated (Sec. III).

Such a bias can be interpreted as the Hawking radiation not being in the thermal state, and hence, one can obtain information from Hawking radiation. However, from a simple thought experiment (Sec. IV.1), we can prove that this interpretation results in a fatal inconsistency.

If it is not interpreted that the bias from the thermal state does not imply the emission of information of the collapsed matter, then either semi-classical quantum field theory is incorrect (Sec. IV.2) or else additional states are accumulated inside the horizon (Sec. IV.3); i.e., a monster is formed. If the monster forms before the Page time, then it is possible to imagine that an unbounded number of states are accumulated inside a black hole, which is theoretically unintuitive.

If a firewall (or non-local interactions) appears before the Page time, it is probable that such a radically new effect can be confirmed or falsified by future astrophysical observations. At the first glimpse, it is fascinating that astrophysical observations can test or falsify some hypothesis of quantum gravity. However, this conclusion also implies that the information loss paradox is indeed worsened, since the paradox appears not only after but also before the Page time.

If our three alternatives are not satisfactory, then the remaining alternative is that information is lost or, at least, effectively lost Lee:2015rwa . We need to know which one is the correct idea. It will be useful to understand the information transfer process not only after but also before the Page time, based on field theory or quantum gravity. Of course, this was beyond the scope of this paper and it was, therefore, left for future research.

DY was supported by a 2-Year Research Grant of Pusan National University. JMB and HZ were supported by the Daegu Gyeongbuk Institute of Science and Technology (DGIST) Undergraduate Group Research Project (UGRP) grant.

Quantum entanglements of a system are generally measured by the entanglement entropy of the system. There are many different definitions of entanglement entropy, one of which is the von Neumann entropy. For a system with the density matrix, $\rho$, the von Neumann entropy is defined by $S(\rho)=-\operatorname{Tr}{\rho\log\rho}$. The log of a matrix can be calculated after diagonalizing the matrix, i.e.,

Then,

To calculate the entanglement entropy of a subsystem, we take partial trace of the density matrix of the total system to obtain the density matrix of the subsystem. The total density matrix is, say, $\rho=\rho_{A}\bigotimes\rho_{B}$; we are interested in the entanglement entropy of $\rho_{A}$. Then,

and the entanglement entropy of $\rho_{A}$ can be calculated same as above, $S(A|B)=-\operatorname{Tr}{\rho_{A}\log(\rho_{A})}$.

In this paper, we considered spin-1/2 systems. For a given $n$-particle system, the state can be expressed by $|{\Psi}_{\textrm{total}}\rangle=\sum_{i_{1}...i_{n}}c_{i_{1}...i_{n}}|i_{1}...i_{n}\rangle,(\text{$i$=1 or 2, where 1 denotes spin up and 2 denotes spin down}).$ The density matrix of the total system is given by $\rho=|\Psi\rangle\langle\Psi|=\sum_{i_{1}...i_{n},j_{1}...j_{n}}c_{i_{1}...i_{n}}c_{j_{1}...j_{n}}^{*}|i_{1}...i_{n}\rangle\langle j_{1}...j_{n}|.$ Then the entanglement entropy of this system can be calculated by following the process described above.

To approximately describe the randomizing process, in Fig. 4, we introduced the swap operation. In doing so, we consider two pairs of maximally entangled particles, namely, $\rho_{a\cup a^{\prime}}$ and $\rho_{b\cup b^{\prime}}$. Therefore, the total number of particles is four.

In the spin-1/2 particle system, we can define

Using the above definitions, we can create two maximally entangled states and their density matrices, $\rho_{a\cup a^{\prime}}$ and $\rho_{b\cup b^{\prime}}$, where

Hence, the total density matrix becomes

With regard to the signs of the spin combinations, there are four possible combinations of the total density matrices; however, there are no significant differences in selecting any combination.

Then, we need to define the swap operator:

where $\mathcal{I}$ is an $2^{4}\times 2^{4}$ identity matrix. For $i^{*},j^{*}\in\left[i_{1},i_{2^{4}}\right]$, the elements of $SW_{\rho}$ become

The physical meaning of this operation is to select and exchange an arbitrary row or column ($i^{*}$ and $j^{*}$, respectively). Since we have defined a swap operator, we can use it to mix our total density matrix, $\rho$.